Backward errors and small-sample condition estimation for ⋆-Sylveter equations

ABSTRACT

In this paper, we consider the perturbation analysis for ⋆-Sylvester equations. Based on the small-sample condition estimation (SCE) technique, we devise algorithms to estimate normwise, mixed and componentwise condition numbers for ⋆-Sylvester equations. We also define a componentwise backward error with a sharp and easily computable bound. Numerical examples illustrate that our algorithm for componentwise perturbations produces reliable estimates, and the new derived computable bound for the componentwise backward error is sharp and reliable for well-conditioned and moderate ill-conditioned ⋆-Sylvester equations under large or small perturbations.

KEYWORDS:

• ⋆-Sylvester equation
• condition number
• componentwise perturbation
• backward error
• small-sample condition estimation

AMS CLASSIFICATIONS:

• 15A09
• 15A12
• 65F35

Acknowledgments

The first author thanks Dr Pengpeng Xie for his comments on the earlier version of this paper, which improve the presentation of this paper.

Disclosure statement

No potential conflict of interest was reported by the authors.